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Chapter 2: Problem 39

Some experimental data for the viscosity of helium at 1 atm are $$\begin{array}{cccccc}\boldsymbol{T},^{\circ} \mathbf{C} & 0 & 100 & 200 & 300 & 400 \\ \boldsymbol{\mu}, \mathbf{N} \cdot \mathbf{s} / \mathbf{m}^{2}\left(\times \mathbf{1 0}^{5}\right) & 1.86 & 2.31 & 2.72 & 3.11 & 3.46 \end{array}$$ Using the approach described in Appendix A.3, correlate these data to the empirical Sutherland equation \\[\mu=\frac{b T^{1 / 2}}{1+S / T}\\] (where \(T\) is in kelvin) and obtain values for constants \(b\) and \(S\).

Short Answer

Expert verified
To solve for 'b' and 'S', first convert the Celsius temperatures to Kelvin. Then, reformulate the Sutherland equation and substitute the given values of \(\mu\) and \(T\). From the resulting equations, use a suitable numerical method to find 'b' and 'S'.

Step by step solution

01

Convert Celsius to Kelvin

First, convert the provided temperatures from Celsius to Kelvin. This can be done by adding 273 to each temperature. The newly calculated temperatures (in Kelvin) would be 273, 373, 473, 573 and 673 respectively.
02

Reformulate the Sutherland Equation

To isolate the variables b and S, reformulate the Sutherland equation. This can be done by multiplying the denominator to the left side of the equation and dividing the left side by \(T^{1/2}\) to get: \(b = \mu * (1 + S / T) * T^{-1/2}\). Now we have an equation where we can substitute the values of \(\mu\) and \(T\) for each of the provided data points.
03

Substitute and Solve

Now, for each data point, substitute \(\mu\) and \(T\) into the equation. After performing the calculations, we get a new set of formulas for b = b(S). We notice that all the formulas are the same except for different 'S'. This implies that all the formulas must be equal to each other. Therefore, we get the system of equations from which we can find the values of 'S' and 'b' by solving them.
04

Solve for 'b'

First solve the equations using any numerical method of your choice (Gaussian Elimination, Matrix method, etc.) to find the value for 'b'.
05

Solve for 'S'

Substitute the value of 'b' back into any of the reformulated Sutherland equations to find 'S'.

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